[Paper Review] Two-sided linear chance constraints and extensions

Miles Lubin, Daniel Bienstock+1 more|arXiv (Cornell University)|Jul 8, 2015

Risk and Portfolio Optimization16 references16 citations

TL;DR

This paper establishes the convexity of two-sided linear chance constraints under Gaussian uncertainty and introduces a second-order cone outer approximation with provably small error, enabling tractable solution of complex nonlinear chance constraints—particularly quadratic ones relevant to power systems. The approach enables practical large-scale optimization under uncertainty with strong theoretical guarantees.

ABSTRACT

We examine the convexity and tractability of the two-sided linear chance constraint model under Gaussian uncertainty. We show that these constraints can be applied directly to model a larger class of nonlinear chance constraints as well as provide a reasonable approximation for a challenging class of quadratic chance constraints of direct interest for applications in power systems. With a view towards practical computations, we develop a second-order cone outer approximation of the two-sided chance constraint with provably small approximation error.

Motivation & Objective

• To establish convexity of two-sided linear chance constraints under Gaussian uncertainty, extending the tractability of chance-constrained optimization.
• To develop a practical, computationally efficient approximation method for quadratic chance constraints arising in power system operations.
• To provide a provably accurate second-order cone relaxation of two-sided chance constraints for use in large-scale optimization.
• To demonstrate the effectiveness of the proposed approximation on real-world power system problems involving renewable energy uncertainty.
• To explore the convexity and tractability of more general nonlinear chance constraints through the lens of two-sided formulations.

Proposed method

• Proves convexity of the two-sided chance constraint $\mathbb{P}(a \leq x^T\xi \leq b) \geq 1 - \epsilon$ for $\epsilon \leq 1/2$ using geometric insights and log-concavity of the Gaussian distribution.
• Develops a second-order cone (SOC) outer approximation of the two-sided chance constraint with provable approximation error bounds.
• Uses the two-sided constraint as a building block to approximate the more complex quadratic chance constraint $\mathbb{P}((a^T\xi + b)^2 + (c^T\xi + d)^2 \leq k) \geq 1 - \epsilon$.
• Applies Schur complement and semidefinite programming techniques to derive tractable SOC formulations for the approximation.
• Compares the proposed SOC approximation with robust and CVaR-based approximations, evaluating performance via numerical integration and visualization.
• Employs a distributionally robust model in Section 4.2 to extend applicability beyond Gaussian assumptions.

Experimental results

Research questions

• RQ1 Is the two-sided linear chance constraint $\mathbb{P}(a \leq x^T\xi \leq b) \geq 1 - \epsilon$ convex in $a$, $b$, and $x$ under Gaussian uncertainty?
• RQ2 Can the two-sided chance constraint be used to construct a tractable and accurate approximation for quadratic chance constraints in power systems?
• RQ3 What is the approximation quality of the proposed second-order cone formulation for two-sided chance constraints, and how does it compare to robust and CVaR-based methods?
• RQ4 Under what conditions is the feasible set of a quadratic chance constraint convex?
• RQ5 Can the proposed method be extended to non-Gaussian elliptical log-concave distributions?

Key findings

• The two-sided chance constraint $\mathbb{P}(a \leq x^T\xi \leq b) \geq 1 - \epsilon$ is convex in $a$, $b$, and $x$ for $\epsilon \leq 1/2$, a result proven using geometric and log-concavity arguments.
• The proposed second-order cone approximation of the two-sided chance constraint has provably small approximation error, enabling reliable use in large-scale optimization.
• For the case $n=2$, the robust approximation using a chi-distribution-based uncertainty set is minimal and cannot be made less conservative without violating feasibility.
• The two-sided approximation yields a simple ball constraint $x^2 + y^2 \leq 1/\Phi^{-1}(1 - \epsilon/4)^2$, which is tighter than the robust box constraint in the $n=2$ case.
• Numerical results show that for $\epsilon = 0.5$, both the robust and two-sided approximations dominate the CVaR approximation; for $\epsilon = 0.05$, no approximation strictly dominates another, and the exact set appears convex.
• The paper conjectures that the feasible set of the quadratic chance constraint is convex for sufficiently small $\epsilon$, based on computational experiments.